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XIX. A Tenth Memoir on Qualities. 
By A. Cayley, F.R.S., Sadlerian Professor of Pure Mathematics in the 
University of Cambridge. 
Received June 12. — Read June 20, 1878. 
The present Memoir, which relates to the binary quintic ( * fx, yf, has been in 
hand for a considerable time : the chief subject-matter was intended to be the theory 
of a canonical form discovered by myself and which is briefly noticed in Salmon’s 
‘Higher Algebra,’ 3rd Ed. (1876), pp. 217, 218; writing a, b, c, d, e, f g . . . u, v, iv 
to denote the 23 covariants of the quintic, then a, b, c, d, f are connected by the 
relation f 2 — — edd fi- arbe — 4c 3 ; and the form contains these co variants thus connected 
together, and also e ; it, in fact, is (1, 0, c, f orb — 3c 3 , are — Af'fx, v/) 5 . 
But the whole plan of the Memoir was changed by Sylvester’s discovery of what 
I term the Numerical Generating Function (N.G.F.) of the covariants of the quintic, 
and my own subsequent establishment of the Beal Generating Function (B.G.F.) of 
the same covariants. The effect of this was to enable me to establish for any given 
degree in the coefficients and order in the variables, or as it is convenient to express 
it, for any given deg-order whatever, a selected system of powers and products of the 
covariants, say a system of “segregates these are asyzygetic, that is, not connected 
together by any linear equation with numerical coefficients ; and they are also 
such that every other combination of covariants of the same deg-order, say every 
“ congregate ” of the same deg-order, can be expressed (and that, obviously, in one 
way only) as a linear function, with numerical coefficients, of the segregates of that 
deg-order. The number of congregates of a given deg-order is precisely equal to 
the number of the independent syzygies of the same deg-order, so that these syzygies 
give in effect the congregates in terms of the segregates : and the proper form in 
which to exhibit the syzygies is thus to make each of them give a single congregate 
in terms of the segregates ; viz., the left hand side can always be taken to be a 
monomial congregate ofif ... or, to avoid fractions, a numerical multiple of such 
form ; and the right hand side will then be a linear function, with numerical coefficients, 
of the segregates of the same deg-order. Supposing such a system of syzygies 
obtained for a given deg-order, any covariant function (rational and integral function 
of covariants) is at once expressible as a linear function of the segregates of that 
deg-order : it is in fact only necessary to substitute therein for every monomial 
congregate its value as a linear function of the segregates. Using the word co variant 
